What are models for? Most developers and users of models, in my experience, seem to assume the answer to this question is obvious and thus never raise it. In fact, modeling has many potential purposes, and some of these conflict with one another. Some of the criticisms made of particular models arise from mis-understandings or mis-perceptions of the purposes of those models, and the modeling activities which led to them.

Liking cladistics as I do, I thought it useful to list all the potential purposes of models and modeling. The only discussion that considers this topic that I know is a brief discussion by game theorist Ariel Rubinstein in an appendix to a book on modeling rational behaviour (Rubinstein 1998). Rubinstein considers several alternative purposes for economic modeling, but ignores many others. My list is as follows (to be expanded and annotated in due course):

1. To better understand some real phenomena or existing system. This is perhaps the most commonly perceived purpose of modeling, in the sciences and the social sciences.

2. To predict (some properties of) some real phenomena or existing system. A model aiming to predict some domain may be successful without aiding our understanding of the domain at all. Isaac Newton’s model of the motion of planets, for example, was predictive but not explanatory. I understand that physicist David Deutsch argues that predictive ability is not an end of scientific modeling but a means, since it is how we assess and compare alternative models of the same phenomena. This is wrong on both counts: prediction IS an end of much modeling activity (especially in business strategy and public policy domains), and it not the only means we use to assess models. Indeed, for many modeling activities, calibration and prediction are problematic, and so predictive capability may not even be possible as a form of model assessment.

3. To manage or control (some properties of) some real phenomena or existing system.

4. To better understand a model of some real phenomena or existing system. Arguably, most of economic theorizing and modeling falls into this category, and Rubinstein’s preferred purpose is this type. Macro-economic models, if they are calibrated at all, are calibrated against artificial, human-defined, variables such as employment, GDP and inflation, variables which may themselves bear a tenuous and dynamic relationship to any underlying economic reality. Micro-economic models, if they are calibrated at all, are often calibrated with stylized facts, abstractions and simplifications of reality which economists have come to regard as representative of the domain in question. In other words, economic models are not not usually calibrated against reality directly, but against other models of reality. Similarly, large parts of contemporary mathematical physics (such as string theory and brane theory) have no access to any physical phenomena other than via the mathematical model itself: our only means of apprehension of vibrating strings in inaccessible dimensions beyond the four we live in, for instance, is through the mathematics of string theory. In this light, it seems nonsense to talk about the effectiveness, reasonable or otherwise, of mathematics in modeling reality, since how we could tell?

5. To predict (some properties of) a model of some real phenomena or existing system.

6. To better understand, predict or manage some intended (not-yet-existing) artificial system, so to guide its design and development. Understanding a system that does not yet exist is qualitatively different to understanding an existing domain or system, because the possibility of calibration is often absent and because the model may act to define the limits and possibilities of subsequent design actions on the artificial system. The use of speech act theory (a model of natural human language) for the design of artificial machine-to-machine languages, or the use of economic game theory (a mathematical model of a stylized conceptual model of particular micro-economic realities) for the design of online auction sites are examples here. The modeling activity can even be performative, helping to create the reality it may purport to describe, as in the case of the Black-Scholes model of options pricing.

7. To provide a locus for discussion between relevant stakeholders in some business or public policy domain. Most large-scale business planning models have this purpose within companies, particularly when multiple partners are involved. Likewise, models of major public policy issues, such as epidemics, have this function. In many complex domains, such as those in public health, models provide a means to tame and domesticate the complexity of the domain. This helps stakeholders to jointly consider concepts, data, dynamics, policy options, and assessment of potential consequences of policy options, all of which may need to be socially constructed.

8. To provide a means for identification, articulation and potentially resolution of trade-offs and their consequences in some business or public policy domain. This is the case, for example, with models of public health risk assessment of chemicals or new products by environmental protection agencies, and models of epidemics deployed by government health authorities.

9. To enable rigorous and justified thinking about the assumptions and their relationships to one another in modeling some domain. Business planning models usually serve this purpose. They may be used to inform actions, both to eliminate or mitigate negative consequences and to enhance positive consequences, as in retroflexive decision making.

10. To enable a means of assessment of managerial competencies of the people undertaking the modeling activity. Investors in start-ups know that the business plans of the company founders are likely to be out of date very quickly. The function of such business plans is not to model reality accurately, but to force rigorous thinking about the domain, and to provide a means by which potential investors can challenge the assumptions and thinking of management as way of probing the managerial competence of those managers. Business planning can thus be seen to be a form of epideictic argument, where arguments are assessed on their form rather than their content, as I have argued here.

11. As a means of play, to enable the exercise of human intelligence, ingenuity and creativity, in developing and exploring the properties of models themselves. This purpose is true of that human activity known as doing pure mathematics, and perhaps of most of that academic activity known as doing mathematical economics. As I have argued before, mathematical economics is closer to theology than to the modeling undertaken in the natural sciences. I see nothing wrong with this being a purpose of modeling, although it would be nice if academic economists were honest enough to admit that their use of public funds was primarily in pursuit of private pleasures, and any wider social benefits from their modeling activities were incidental.

POSTSCRIPT (Added 2011-06-17): I have just seen Joshua Epstein’s 2008 discussion of the purposes of modeling in science and social science. Epstein lists 17 reasons to build explicit models (in his words, although I have added the label “0” to his first reason):

Belately, I want to record a play seen at the headquarters of The Royal Society in London last month, Let Newton Be, written by Craig Baxter, but using only Isaac Newton’s own words. The play was interesting although the energy of the play sagged at times, particularly in the first half. The story only barely mentioned Newton’s interest in alchemy, and seemed to overlook his brutal, deadly campaigns against money forgers later in life (or did I nap through that scene?)

The play comprised three actors, two men and a woman, who played Newton at different ages – as a child, as a young-ish Cambridge academic, and as an old man. As a work of drama, the conceit worked well, although it was best when one of the actors was playing another person interacting with Newton (eg, Halley, and later Leibniz, who spoke in an amusing cod-German accent). Perhaps the real Newton was not sufficiently schizoid for three actors to play him, at least not when constrained to only use the man’s written words. As I have remarked before, Newton’s personality was all of a piece: it is only modern westerners who cannot imagine a religious motivation for activities such as scientific research, for example, or who find alchemy and calculus incoherent.

The performance was followed by a panel discussion by the Great and the Good – two historians and two scientists. One of the scientists was the Astronomer Royal, Sir Martin Rees, who has subsequently won this year’s Templeton Prize for Science and Religion. The discussion was interesting, so it is a pity it was not recorded for posterity.

A review of another play about a member of the matherati, Kurt Godel, is here.

Another concert caught in Bologna this past weekend was a short concert for Palm Sunday in the Crypt of the Basilica of San Pietro. The music was by Quartetto d’Archi G. B. Martini and Corale Convivium Musicum, playing the following programme:

The acoustics of the crypt were surprisingly good: despite the stone walls and columns, the low, curved ceilings bounced the sound around the chamber, and the crowd absorbed it well. Perhaps all the palm fronds being waved helped. The music was performed well, although the concert was over in under 30 minutes. We were left wanting more.

A superb concert last night in Bologna, with Orchestra Mozart and the Mahler Chamber Orchestra combining forces under young conductor Diego Matheuz. The concert took place in Auditorium Manzoni, where I have enjoyed concerts before, sometimes under Maestro Abbado. This hall has a relatively modern interior, almost fan-shaped, with undulating wooden walls and an undulating wooden ceiling over the stage. The acoustic is warm, bright and fast. The stage is only small, and barely took the forces arranged last night. The cellos were placed in the middle, with the violas on the conductor’s immediate right, and so the sound of the violas may well have been lost. Similarly, only the percussion and brass were (slightly) elevated, the woodwinds seated at the same level as the strings. I was close enough not to miss anything from these placements.

Prokofiev’s First Violin Concerto was played by Vadim Repin, who also played Ravel’s violin rhapsody, Tzigane. Both pieces were fiery and technically impressive, my strong distaste for Prokofiev’s music notwithstanding. His music strikes me as truly incoherent, using types of expression (eg, multiple simultaneous keys) and modes of musical cognition that are alien to me. My distaste is stronger than mere dislike, being incomprehension. The abrupt change in mood, for example, between the second and third movements, seems meant to provoke the listener, as if to say, I have the power to change your attitude to this music at a whim, and to prove it, I will now do it. Who could enjoy the company of such a person?

I have heard Repin perform before, a few years ago in Barcelona (playing the Sibelius concerto). As on that occasion, he encored with theme and variations of Carnival of Venice, a crowd-stopping showpiece of skill and effects made famous for violinists by Pagannini and for trumpeters by Arban. This time, however, Repin began with a fiery introduction, then detoured into several bars of accompaniment vamping before launching the theme. The vamping allowed him to signal to the orchestral musicians what to play as they joined him, something he had tried unsuccessfully in Barcelona while himself playing the theme.

The concert also included Stravinsky’s Firebird Suite, in what was certainly the most thrilling, spine-tingling, edge-of-seat performance of this work I have ever heard. Matheuz conducted from memory, which is not nothing for this jagged music, and his energy and enthusiasm was compelling. The principal violinists had swapped places for this piece. Before the interval, the principal for the Mahler CO, Gregory Ahss, was lead. For the Stravinsky after interval, Orchestra Mozart’s principal, Raphael Christ, took over. I was seated close enough to see them play, and both were very impressive. Both people to watch, along with Matheuz.

The Auditorium Manzoni is mildly fan-shaped, a shape that is not common for concert halls. (Another example is the art deco Philharmonic Hall in Liverpool, UK, whose fan shape is much more pronounced.) The walls around the stage and the hall, along with the ceiling over the stage have an undulating wooden veneer, which would help sound propagation in diverse directions. The balcony overhands a large part of the auditorium, but at quite a high level, so that seats under the balcony are not “dark” in terms of the sounds they receive from the stage.

The photo shows Claudio Abbado in London in October 2011 with the Lucerne Festival Orchestra, which shares members with the ensemble seen in Bologna . Credit: Chris Christodoulou.

Alina Ibragimova – Mendelssohn’s E minor Concerto, with the Orchestra of the Age of Enlightenment (London, 2012); Schumann’s Concerto, with the London Symphony Orchestra (London, 2014). Schumann wrote his concerto for Joachim (pictured), who never performed it publicly, and tried to keep it out of print for a long time. Pity that Joachim did not succeed.

The drawing is Adoph Menzel’s 1853 drawing of Joseph Joachim (1831-1907), famously a pupil of Mendelssohn and a cousin of Ludwig Wittgenstein’s grandmother. Joachim taught Jeno Hubay (1858-1937), who taught Leo Birsen (1902-1992), with whom I had some lessons.

Writing in Bertinoro, Italy, I have just learnt of the death earlier this year of J. Martin Harvey (1949-2011), a friend and former colleague, and one of Zimbabwe’s great mathematicians. Martin was the first black student to gain First Class Honours in Mathematics from the University of Rhodesia (as it then was, now the University of Zimbabwe), the first person to gain a doctorate in mathematics from that University, and the first black lecturer appointed to teach mathematics there. (Indeed, his three degree certificates name three different universities – Rhodesia, London, and Zimbabwe – but all were granted by the same physical institution.) He later became an actuary, one of the few of any colour in Zimbabwe, but this was a career that lost value with the declining Zimbabwe dollar: actuarial science is about financial planning under uncertainty, and planning is impossible and pointless in an economy with hyper-inflation. He then became part of the great Zimbabwean diaspora, lecturing at the University of the Western Cape, in South Africa.

Martin was a true child of the sixties, with all the best qualities of that generation – open, generous, tolerant, curious, unpompous, democratic, sincere. He was a category theorist, and like most, a deep thinker. Martin was a superb jazz flautist, and on his travels would seek out jazz musicians to jam with. He wrote and recited poetry, and indeed could talk with knowledge on a thousand topics. I once spent a month traveling the country with him on a market resarch project we did together, and his conversation was endlessly fascinating. Despite our very different childhoods, I recall a long, enjoyable evening with him in a shebeen in rural Zimbabwe talking about the various American and Japanese TV series we had both seen growing up (which I mentioned here). Among many memories, I recall him once arguing that a university in a marxist state should have only two faculties: a Faculty for the Forces of Production, and a Faculty for the Relations of Production. There was great laughter as he insisted that the arts and humanities were essential to effective production, and so belonged in the former faculty; this argument was typical of his wit and erudition.

Our mutual friend, Heneri Dzinotyiweyi, another great Zimbabwean mathematician, has a tribute here. I send my condolences to his wife, Winnie Harvey, and family. Vale, Martin. It has been an honour to have known you.

The image shows Container Series (2009), a ceramic installation by contemporary Chinese artist Liu JianHua, recently acquired by the Art Gallery of New South Wales. From the description of the work by the AGNSW:

In this beautiful celadon work ‘Container Series’ 2009, Liu acknowledges the magnificent ceramic heritage of China by re-creating traditional ritual vessel shapes such as stem cups and ‘gu’ ( wine vessel), and placing them alongside more contemporary shapes to present an impressive installation of 37 pieces of varying size and shape. The pieces are randomly arranged, in contrast to the formal presentation of temple and imperial settings of the past. Arranged as an installation they are placed as on a blank canvas. Liu has used a celadon glaze, in emulation of the classic greenwares of the Song dynasty (960-1279). All the pieces look to be filled with a red liquid, but the ceramics are in fact hollow, with a red glaze on top of each piece giving the illusion of liquid. Like the celadon, the red is another classic of the Chinese ceramic repertoire, traditionally known as ‘sang de boeuf’ or oxblood (‘langyao hong’). The way the artist has used the two colours to dramatically contrasting effect is innovative and effective. The red colour evokes blood, and the work may be a homage to those whose blood has been spilt in the pursuit of specific goals. Each piece is handmade and thrown on a wheel. They were glazed first with the celadon then the red ‘langyao hong’ glaze, after which they were fired only once in a kiln at about 1342 degrees centigrade. Most parts were made in varying numbers of editions.

Eugene Tan has noted of Liu’s work that it, ‘reflects the complex ontological relation between the production of consumer goods in China and the international art system, thereby also reflecting the recent, rapid growth of the Chinese contemporary art market.’ (Tan, undated) As such, Liu’s work makes allusions to the present situation in China both culturally and economically.”

[Note: The commentary by Eugene Tan is dated 2009-07-02 on the web-page linked to here.]

A description of another installation of this artist can be found here.

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